Navier-Stokes

The Navier-Stokes regularity problem—whether smooth solutions in three dimensions develop singularities—remains one of the most persistent challenges in mathematical physics. This paper proposes that the problem is not resolved by finding new bounds on R3 (Euclidean space), but by identifying the correct compact micro-geometry of physical space. Within the S3-MEP framework (compact S3 ~ SU(2) geometry, one-bit information pixel, Maximum Caliber dynamics), we establish a rigorous uniqueness result: the 3-sphere is the sole compact connected orientable 3-manifold capable of sustaining a physically consistent fluid dynamics.

Our argument proceeds by an exhaustive elimination operating on the entire landscape of Thurston geometrization. We impose two criteria: (G1) Bochner-coercivity (Ric > 0) and (G2) an integer spectral budget compatible with the electroweak interaction window.

• Criterion G1 eliminates the seven non-spherical Thurston geometries (including the infinite families of hyperbolic, flat, and Seifert-fibered manifolds) via curvature constraints.

• Criterion G2 eliminates all non-trivial spherical space forms (including the Poincaré homology sphere and infinite families of lens spaces) via spectral rarefaction, which pushes the electroweak scale to physically inadmissible values (10^-25 GeV or 10^19 GeV).